The Aventine is a great suggestion
I strongly second JBD's suggestion of the Aventine. In fact, I am currently writing this from a lovely apartment on Viale Manlio Gelsomini. I found the apartment on Airbnb, but I also looked and VRBO and FlipKey. Although I didn't bring a car, the apartment rental comes with space for one and there is actually legal street parking on this block as well.
The Aventine is great location for a family-- I am here escorting my older daughter to boarding school (St. Stephens) and my 12 year old daughter is also with me. There are many Italian and international families in the area, I see strollers and dogs out with people at all hours.
The neighborhood is, in many ways, reminescent of lower manhattan in the mid eighties. That is, pricey rents, but a lot of graffitti and a very diverse, international flavor. Several good places to eat within a 10 minute walk. The roman transit system is fast and easy to navigate as well.